Changes in the Sea-level. 205 



The second part of our integral, that arising from the sub- 

 traction of the force at the centre of figure, may be deduced from 

 this by observing that if we suppose the mass dm removed in- 

 definitely backwards on the same radius, and at the same time 

 increased in magnitude so as to preserve its attraction on the 



centre (—J the same throughout, we shall approach indefi- 

 nitely to the state in which every part of the earth is ' solicited 

 by equal and parallel attractions of the required magnitude. 

 But by this process all the terms of the series (1), after the first, 

 vanish. The first term, therefore, is the quantity sought to be 

 subtracted, and the first and second parts of our integral united, 

 as due to dm, become 



_ 5 _jP 2+ P 3 -+P 4 - ? + &cj 



(2) 



To obtain what is due to the whole action of the ice, we must 

 express dm in terms of /// and co', and integrate from co 1 = to 

 &/ = 27r, and from jul' = /n l to /// = 1, f*> x being the extreme value of ///. 

 Now dm is the volume of an elementary prism of length /3, 

 and whose base is s sin d'dco' x sd6'; or, as sin 6'd6'= — dfjJ, it is 

 fis^d/ju'dco 1 , if we take the integration in the direction in which jj! 

 increases. Substituting this value for dm, and integrating first 

 in co', all the terms in (co — co') obviously vanish, and the others 

 take 27T as a factor, so that we get 



Now* 

 whence 



—^■ydf! {Q' 2 Q 2 + Q' 3 Q 3 ^- + &c} 



n , 1 f dQ! n+1 rfQ',-i \ 



Hn ~2n + l\ dfj dp' J' 



JQf^C + gjL- {Q'„ +1 -QC}; 



and when ft'=l, Q'n+i = Qn-i = l- Therefore between the 

 limits the integral is 



fr> (1) n (1)1 1 



2n+l ^«-i~Vi)' 

 which we will call A n ; and then the part of the integral 



* Bertrand, as above. Murphy also, in his ' Electricity,' p. 50, gives 

 the equivalent integral. 



